A386922 Denominators of the partial sums of 1/d(prime(k)+1), where d is the number of divisors function.

2, 6, 12, 3, 2, 4, 12, 12, 24, 3, 2, 4, 8, 24, 120, 15, 20, 5, 30, 20, 10, 5, 60, 30, 15, 120, 60, 15, 120, 60, 120, 40, 20, 15, 60, 120, 120, 120, 240, 240, 720, 720, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 1008, 1008, 1008, 1008, 63, 1008, 5040, 5040

Offset

1,1

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Mikhail R. Gabdullin, Vitalii V. Iudelevich, and Sergei V. Konyagin, Karatsuba's divisor problem and related questions, arXiv:2304.04805 [math.NT], 2023.

Vitalii V. Iudelevich, On the Karatsuba divisor problem, Izvestiya: Mathematics, Vol. 86, No. 5 (2022), pp. 992-1019; arXiv preprint, arXiv:2304.03049 [math.NT], 2023.

Formula

a(n) = denominator(Sum_{k=1..n} 1/A008329(k)).

Example

Fractions begin with 1/2, 5/6, 13/12, 4/3, 3/2, 7/4, 23/12, 25/12, 53/24, 7/3, 5/2, 11/4, ...

Mathematica

Denominator[Accumulate[1/DivisorSigma[0, Prime[Range[100]] + 1]]]

Prog

(PARI) list(lim) = {my(s = 0); forprime(p = 1, lim, s += (1/numdiv(p+1)); print1(denominator(s), ", ")); }

Crossrefs

Cf. A000005, A008329, A008864, A104529, A386921 (numerators).

Sequence in context: A293122 A014452 A188894 * A355416 A054579 A277962

Adjacent sequences: A386919 A386920 A386921 * A386923 A386924 A386925

Keyword

nonn,frac,easy

Author

Amiram Eldar, Aug 08 2025

Status

approved